ch9-5 - SECTION 9.9 Castigliano's Theorem 601 Castigliano's...

Castigliano’s TheoremThe beams described in the problems for Section 9.9 have constant flexural rigidity EI.Problem 9.9-1A simple beamAB of length Lis loaded at the left-handend by a couple of moment M0(see figure). Determine the angle of rotation uAat support A. (Obtain the solutionby determining the strain energy of the beam and then using Castigliano’stheorem.) Solution 9.9-1Simple beam with couple M0ABLM0SECTION 9.9Castigliano’s Theorem601(downward)5M0¢12xL≤M5M02RAx5M02M0xLRA5M0LSTRAIN ENERGYCASTIGLIANO’S THEOREM(clockwise)(This result agree with Case 7, Table G-2)uA5dUdM05M0L3EIU5#M2dx2EI5M202EI#L0¢12xL≤2dx5M20L6EIABLM0xProblem 9.9-2The simple beam shown in the figure supports aconcentrated loadPacting at distance afrom the left-hand support anddistanceb from the right-hand support. Determine the deflection dDat point Dwhere the load is applied.(Obtain the solution by determining the strain energy of the beam andthen using Castigliano’s theorem.) Solution 9.9-2Simple beam with load PABDLabPMDB5RBx5PaxLMAD5RAx5PbxLRB5PaLRA5PbLSTRAIN ENERGYCASTIGLIANO’S THEOREM(downward)dD5dUdP5Pa2b23LEIU5UAD1UDB5P2a2b26LEIUDB512EI#b0¢PaxL≤2dx5P2a2b36EIL2UAD512EI#a0¢PbxL≤2dx5P2a3b26EIL2U5#M2dx2EIDLPxx

This preview
has intentionally blurred sections.
Sign up to view the full version.

Problem 9.9-3An overhanging beamABCsupports a concentrated loadPat the end of the overhang (see figure). SpanAB has length Land theoverhang has length a. Determine the deflection dCat the end of the overhang. (Obtain the solution by determining the strain energy of the beam and then usingCastigliano’s theorem.) Solution 9.9-3Overhanging beam602CHAPTER 9Deflections of BeamsABCLaP(downward)MCB52PxMABRAxPaxLRA5PaLSTRAIN ENERGYCASTIGLIANO’S THEOREM(downward)dC5dUdP5Pa23EI(L1a)U5UAB1UCB5P2a26EI(L1a)UCB512EI#a0(2Px)2dx5P2a36EIUAB512EI#L0¢2PaxL≤2dx5P2a2L6EIU5#M2dx2EICPxxProblem 9.9-4The cantilever beam shown in the figure supports a triangularly distributed load of maximum intensity q0. Determine the deflection dBat the free end B. (Obtain the solution by determining the strain energy of the beam and then using Castigliano’s theorem.) Solution 9.9-4Cantilever beam with triangular loadALBq0P5fictitious load corresponding to deflection dBMPx2q0x36LSTRAIN ENERGYCASTIGLIANO’S THEOREM(downward)(This result agrees with Cases 1 and 8 of Table G-1.)SETP50:dB5q0L430EIdB50U0P5PL33EI1q0L430EI5P2L36EI1Pq0L430EI1q20L542EIU5#M2dx2EI512EI#L0¢2Px2q0x36L≤2dxALBq0Px